A New Type Of Trust Region Method Guided By The Negative Gradient Direction

The trust region method is a kind of efficient numerical value calculating method forunconstrained optimization problems. Solving trust region sub problem is the core part of thetrust region algorithm. In order to obtain better convergence and numerical evaluation results,the centre of sphere and radius of trust region are widely researched by scholars. In this paperwe present a new type of trust region method guided by the negative direction forunconstrained optimization problems.Firstly, in the solution of trust region sub problem within the classical trust regionalgorithms, the centre of sphere is the current interaction point and one step-size is the upperbound. Considering that only with the negative gradient direction to acute angle may reducethe value of the function value, we introduce the parameter to the control of the centre ofsphere and the radius. We prove that the new trust region method is globally convergence.Secondly, trust region algorithms guided by the negative gradient decrease the efficiencywhen the current iteration point is close to the optimum one. In this paper, we propose analgorithm for nonlinear optimization problem that employs both classical trust region methodsand two improved trust region methods. When the current iteration point is close to theoptimum one, we use the traditional trust region method, otherwise use the two improved trustregion method. Numerical evaluation demonstrates the validity of the new trust regionalgorithms.